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Proofs from THE BOOK

Springer-Verlag Berlin Heidelberg GmbH

Martin Aigner Gunter M. Ziegler

Proofs from THE BOOK With 220 Figures Including Illustrations by Karl H. Hofmann

Springer

Martin Aigner

Gi.inter M. Ziegler

Freie Universitat Berlin Institut fiir Mathematik II (WE2) Arnimallee 3 D-14195 Berlin, Germany email: [email protected]

Technische Universitat Berlin Fachbereich Mathematik, MA 7-1 StraBe des 17. Juni 136 D-1 0623 Berlin, Germany email: [email protected]

CIP data applied for Die Deutsche Bibliothek- CIP-Einheitsaufnahme Aigner, Martin: Proofs from the book/Martin Aigner; Gunter M. Ziegler. Incl. ill. by Karl H. Hofmann.Berlin; Heidelberg; New York; Barcelona; Budapest; Hong Kong; London; Milan; Paris; Singapore; Tokyo: Springer, 1998 ISBN 978-3-662-22345-1

Mathematics Subject Classification (1991): 00-01 (General)

ISBN 978-3-662-22343-7 (eBook) ISBN 978-3-662-22345-1 DOI 10.1007/978-3-662-22343-7

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting. reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1998 Originally published by Springer-Verlag Berlin Heidelberg New York in 1998 Softcover reprint of the hardcover I st edition 1998

The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement. that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typeset in eTEX by the authors Computer-to-plate processing by Mercedes- Druck GmbH, Berlin Cover design: de'blik, Berlin SPIN I 0655190

46/3143- 54 3 2 I 0- Printed on acid-free paper

Preface

Paul Erdos liked to talk about The Book, in which God maintains the perfect proofs for mathematical theorems, following the dictum of G. H. Hardy that there is no permanent place for ugly mathematics. Erdos also said that you need not believe in God but, as a mathematician, you should believe in The Book. A few years ago, we suggested to him to write up a first (and very modest) approximation to The Book. He was enthusiastic about the idea and, characteristically, went to work immediately, filling page after page with his suggestions. Our book was supposed to appear in March 1998 as a present to Erdos' 85th birthday. With Paul's unfortunate death in the summer of 1997, he is not listed as a co-author. Instead this book is dedicated to his memory. We have no definition or characterization of what constitutes a proof from The Book: all we offer here is the examples that we have s